Not sure how MP3Tag handles FLAC files, but if it just shows the tags as they are stored it should just display what Picard has written. If it uses some special tag mapping (see below) it maybe does not support it for FLAC files. For MP3 make sure to configure Picard to write ID3 v2.4 as described above.
This is just the different name for that tag used in MP3Tag and Picard. MP3Tag uses uppercase "MOOD", Picard lowercase "mood" (not sure where you got the idea from Picard is writing "Mood").
In general there is not the one "mood" tag. There is the general concept of a "mood" tag, but how this is written to the file depends on the tagging format used. E.g. for ID3 there is a tag called TMOO which is used for this, and in WMA it would be WM/Mood. In order to deal with this Picard has what we call an internal tag name, which is just lowercase "mood", and that internal tag gets mapped onto the format specific tag.
MP3Tag has similar concept documented in http://help.mp3tag.de/main_tags.html . There you can see the MP3Tag internal name in the first column, and it is "MOOD".
Just that I don't use both tools, so I can give just general advice on Picard. Clarified my previous comment a bit